Function in mathematics

https://en.wikipedia.org/wiki/Function_(mathematics)

A function (map, mapping, correspondence, transformation) is a binary relation between two sets that associates to each element of the first set exactly one element of the second set.

A function is a process (rule) that associates each element x of a set X (the domain of the function) to a single element y of a set Y (the codomain of the function), with the possibility that the domain and codomain are the same set.

Functions originally represented how a varying quantity depends on another quantity; e.g. the position of a planet is a function of time. The concept of a function was formalized at the end of the 19th century in terms of set theory.

A function is a relation that is right-unique and left-serial.

This relation is denoted by f (x) = y, where the element x is the input (argument) value to f and y is its output value; it is also said that y is the image of x by f, with x as the pre-image.

The input *argument, M, is bound by a function, f, by a corresponding formal parameter, x, which is its declaration occurrence, while all the places inside the function's body, where that parameter is used, are called its application occurrences.

                                   ┌ function application

┌ function ┐ │ to an │declaration│ │ ┌ argument │ │ │ │ f(x) = x + 2x f(5) = 5 + 2*5 │ │ │ │ parameter │ │ │ │ │ └────┴─ application (x+2x)[5/x] │ │ │ │ occurrences substitution │ │ │ └ parameter declaration occurrence │ └ name of the function

A function, just like a relation, is uniquely represented by a set of ordered pairs, called the graph of the function, f = { ∀x ∈ X. ∃y ∈ Y | (x, y) }.

∀x ∈ X. ∃y ∈ Y. (x, y) ∈ f ∧ y = f(x)